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This theoretical paper sets forth two "aspects of predication," which describe how students perceive the relationship between a property and an object. We argue these are consequential for how students make sense of discrete mathematics proofs related to the properties and how they construct a logical structure. These aspects of predication are (1) "populating" the way students generate sets of examples of the property, and (2) "testing membership" how one tests whether or not a given object has a specific property. Using data from two teaching experiments in which undergraduate students read proofs of theorems about the discrete concept of multiple relations, we illustrate the nature of these aspects of predication and demonstrate how they help explain student interpretations of the proofs. We argue that these particular properties from number theory likely have correlates in many other discrete mathematics topics because of the role of computation/algorithms for defining and testing properties as well as the role of iteration and recursion in populating examples. We anticipate that these constructs will be useful to teachers and researchers of discrete mathematics to foster and assess student understanding of various mathematical properties. They provide tools for thinking about what it means to understand properties in a rich and coherent way that supports understanding complex lines of inference and generalizations.

This study compares the relative influence of syntax, semantics, and pragmatics in university students' interpretation of multiply quantified statements in mathematics, both before and after instruction. Like previous studies, results show that semantics plays a heavy role in student interpretation, especially before instruction. Unlike previous studies, our data suggests that the patterns of student interpretation rely more upon the mathematical context than upon the order of the quantifiers. We operationalize two of Grice's (1975) pragmatic Maxims to evaluate whether they help explain which interpretations are harder for students to adopt for various statements. Our data support the claim that students find it easier to construct relevant interpretations, but do not support the claim that students find it easier to construct interpretations that render the statement true. Finally, based on our sample from six Transition to Proof classes across the US, we observe that after their experiences in such courses students became more sensitive to syntax in their interpretation of the statements.

This paper introduces the concept of logical consistency in students' thinking in mathematical contexts. We present the Logical in-Consistency (LinC) instrument as a valuable assessment tool designed to examine the prevalence and types of logical inconsistencies among undergraduate students' evaluation of mathematical statements and accompanying arguments. Beyond its utility as an assessment tool, this paper also underscores the potential of the LinC instrument as an instructional aid, effective for identifying students who may inadvertently overlook logical contradictions within their mathematical assertions. Furthermore, this paper outlines a practical classroom activity based on the LinC instrument format. This activity is tailored to enhance students' capacity for reasoning with logical consistency, specifically in proof-oriented mathematics courses. We hope this activity helps instructors address and mitigate logical inconsistencies in students' thinking throughout their classroom activities. [ABSTRACT FROM AUTHOR]

The purpose of this study is to explore how an introductory real analysis (IRA) course can be designed to bridge a gap between students' intuition and mathematical rigor. In particular, we focus on a task, called the e-strip activity, designed for the convergence of a sequence. Data were collected from a larger study conducted as a classroom teaching experiment for a semester-long IRA course. Fischbein's notion of secondary intuition was employed to elucidate the development of student intuition throughout the e-strip activity. We discuss how the e-strip activity played a role in developing students' intuition and how it impacted students' learning of the subsequent topics in the course, including definitions and theorems about convergence.

The purpose of this paper is to provide issues related to student understanding of logical components that arise when solving word problems. We designed a logic problem called the King and Prisoner Puzzle--a linguistically simple, yet logically challenging problem. In this paper, we describe various student solutions to the puzzle and discuss the issues with students' logic. In particular, it is thought-provoking that invalid arguments in students' solutions to the puzzle are based on a lack of precise understanding of some basic logical components. This emphasizes the necessity of additional teaching to form mutual understanding of the meaning of the logical components.

In this article, we suggest an instructional intervention to help students understand statements involving multiple quantifiers in logical contexts. We analyze students' misinterpretations of multiple quantifiers related to the epsilon-N definition of convergence and point out that they result from a lack of understanding of the significance of the order of the quantifiers in the definition. We introduce the Mayan activity which is designed to cause and then to help resolve students' cognitive dissonance. In particular, the Mayan stonecutter story in the activity is presented in an understandable and colloquial form so that students can recognize the independence of epsilon from N in the epsilon-N definition. Consequently, the Mayan activity can be regarded as a useful instructional intervention to study statements related to the epsilon-N definition of convergence. (Contains 1 table.)

This study explored students' understanding of a logical structure in defining the limit of a sequence, focusing on the relationship between epsilon and N. The subjects of this study were college students who had already encountered the concept of limit but did not have any experience with rigorous proofs using the epsilon-N definition. This study suggested two statements, each of which is written by using a relationship between epsilon" and N, similar to the epsilon-N definition. By analyzing the students' responses to the validity of the statements as definitions of the limit of a sequence, students' understanding of such a relationship was classified into five major categories. This paper discusses some essential components that students must conceptualize in order properly to understand the relationship between epsilon and N in defining the limit of a sequence.

This article suggests an activity, called the epsilon-strip activity, as an instructional method for conceptualization of the rigorous definition of the limit of a sequence via visualization. The article also describes the learning objectives of each instructional step of the activity, and then provides detailed instructional methods to guide students to reach the learning objectives. (Contains 10 figures and 3 tables.)

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students' understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students' images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the [epsilon]-strip activity. This paper illustrates how these students' understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.